Finite Group Databasebeta

← #13155680

Subgroups — #13155680

Order 1728.

634 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88811C211yesmaximal, characteristic
2C3×#63894C211yesmaximal, characteristic
3C3×#63844C211yesmaximal, characteristic, complemented
4#11722039C211yesmaximal, characteristic, complemented
5#11722033C211yesmaximal, characteristic
6#11722019C211yesmaximal, characteristic, complemented
7C3×#63576C211yesmaximal, characteristic, complemented
8#10590590—31nomaximal
9#10590457—31nomaximal
10#10590358—31nomaximal
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C3×#624—21no—
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×C4⋊Dic3C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C32×C4×Dic3C2211yescharacteristic
19#88811C414yescomplemented
20C3×#3724—21no—
21C3×#3722—21no—
22C3×Dic32—21no—
23C3×#893—31no—
24C3×#887—31no—
25C3×C4×D12—31no—
26C3×C4×Dic6—31no—
27#63894D311yescharacteristic
28#63844D311yescharacteristic, complemented
29C2×#3717—31no—
30C2×#3716—31no—
31C2×#3715—31no—
32#63637—31no—
33#63636—31no—
34#63630—31no—
35C3×#807—31no—
36C3×#807—31no—
37#63576D311yescharacteristic, complemented
38#63565—31no—
39#63565—31no—
40#63565—61no—
41#63562—31no—
42#63562—31no—
43#63562—61no—
44#63554—31no—
45#63554—31no—
46#63554—61no—
47C22×C3×C3⋊D3—21no—
48C2×C33×C4C2311yescharacteristic
49C2×C3×C3⋊Dic3—21no—
50C3×C4⋊(C3⋊D3)—21no—
51C3×C4⋊(C3⋊D3)C2×C412yes—
52C3×#634—21no—
53C3×#634C2×C412yescomplemented
54C2×C32×Dic3—21no—
55C2×C32×Dic3—21no—
56C2×C32×Dic3D411yescharacteristic
57C2×C32×Dic3D411yescharacteristic
58C3×#624—42no—
59C3⋊(C3⋊(C3⋊C8))—21no—
60C3×C3⋊(C3⋊C8)C2×C412yescomplemented
61#5026—91no—
62#5018—91no—
63#4889—91no—
64C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
65C2×#634C2×D311yescharacteristic
66C2×C3×D12—31no—
67C2×C3×D12—31no—
68C2×C3×D12—61no—
69C2×C3×C4×D3—61no—
70C2×C3×C4×D3—61no—
71C2×C3×Dic6—31no—
72C2×C3×Dic6—31no—
73C2×C3×Dic6—61no—
74C2×#624—21no—
75C32×C4⋊C4—31no—
76C32×C4⋊C4—31no—
77C32×C42—31no—
78C2×C3⋊(C3⋊C8)—31no—
79C2×C3⋊(C3⋊C8)—31no—
80C2×C3⋊(C3⋊C8)—31no—
81C2×C3⋊(C3⋊C8)—61no—
82C2×C3⋊(C3⋊C8)—61no—
83C2×C3⋊(C3⋊C8)—61no—
84C2×C3⋊(C3⋊C8)—121no—
85C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
86C3×(C2×D3)⋊C4—61no—
87C3×(C2×D3)⋊C4—61no—
88C3×C4⋊Dic3—21no—
89C3×C4⋊Dic3—21no—
90C3×C4⋊Dic3C2×D311yescharacteristic
91C3×C4⋊Dic3C2×D311yescharacteristic
92C3×C4⋊Dic3C2×D311yescharacteristic
93C3×C4⋊Dic3C2×D311yescharacteristic
94C3×Dic3⋊C4—61no—
95C3×Dic3⋊C4—61no—
96C3×C4×Dic3—21no—
97C3×C4×Dic3—61no—
98C3×C4×Dic3—61no—
99C3×C4×Dic3C2×D311yescharacteristic
100C3×C4×Dic3C2×D311yescharacteristic
101C2×C3×C3⋊C8—31no—
102C2×C3×C3⋊C8—31no—
103C2×C3×C3⋊C8—61no—
104#3724—21no—
105#3722—21no—
106Dic32—21no—
107#3717—34no—
108#3716—34no—
109#3715—34no—
110C22×C33C2×D411yescharacteristic
111C2×C3×C3⋊D3—22no—
112C2×C3×C3⋊D3—41no—
113C2×C3×C3⋊D3—41no—
114C33×C4C22×C411yescharacteristic
115C33×C4#4111yescharacteristic
116C3×C3⋊Dic3—22no—
117C32×Dic3—21no—
118C32×Dic3—21no—
119C32×Dic3—41no—
120C32×Dic3—41no—
121C3×C4×Q8—91no—
122C3×C4×D4—91no—
123C2×#215—91no—
124C2×#214—91no—
125C2×#204—91no—
126#893—31no—
127#887—31no—
128C4×D12—31no—
129C4×Dic6—31no—
130C3×#107—91no—
131#835—91no—
132#832—91no—
133#815—91no—
134#810—91no—
135#810—91no—
136#810—181no—
137#809—91no—
138#809—91no—
139#809—181no—
140#807—31no—
141#807—31no—
142#803—91no—
143#803—91no—
144#803—181no—
145C22×C3⋊D3—21no—
146C22×C3×D3—61no—
147C22×C3×D3—61no—
148C22×C3×D3—121no—
149C2×C32×C4—21no—
150C2×C32×C4—21no—
151C2×C32×C4—21no—
152C2×C32×C4—31no—
153C2×C32×C4—31no—
154C2×C32×C4—41no—
155C2×C32×C4—61no—
156C2×C32×C4—61no—
157C2×C32×C4C22×D311yescharacteristic
158C2×C32×C4C22×D311yescharacteristic
159C2×C32×C4C22×D311yescharacteristic
160C2×C3⋊Dic3—21no—
161C4⋊(C3⋊D3)—21no—
162C4⋊(C3⋊D3)C4×D312yes—
163#634—21no—
164#634C4×D312yescomplemented
165C2×C3×Dic3—21no—
166C2×C3×Dic3—21no—
167C2×C3×Dic3—21no—
168C2×C3×Dic3—21no—
169C2×C3×Dic3—21no—
170C2×C3×Dic3—21no—
171C2×C3×Dic3—41no—
172C2×C3×Dic3—41no—
173C2×C3×Dic3—61no—
174C2×C3×Dic3—61no—
175C2×C3×Dic3—121no—
176C2×C3×Dic3#6711yescharacteristic
177C2×C3×Dic3#6711yescharacteristic
178C2×C3×Dic3#6711yescharacteristic
179C2×C3×Dic3#6711yescharacteristic
180C3×D12—32no—
181C3×D12—32no—
182C3×D12—62no—
183C3×D12—61no—
184C3×D12—61no—
185C3×D12—121no—
186C3×C4×D3—122no—
187C3×C4×D3—122no—
188C3×Dic6—32no—
189C3×Dic6—32no—
190C3×Dic6—62no—
191C3×Dic6—61no—
192C3×Dic6—61no—
193C3×Dic6—121no—
194#624—42no—
195C3⋊(C3⋊C8)—61no—
196C3⋊(C3⋊C8)—62no—
197C3⋊(C3⋊C8)—61no—
198C3⋊(C3⋊C8)—61no—
199C3⋊(C3⋊C8)—121no—
200C3⋊(C3⋊C8)—121no—
201C3⋊(C3⋊C8)—122no—
202C3⋊(C3⋊C8)C4×D312yescomplemented
203C3×C3⋊C8—32no—
204C3×C3⋊C8—32no—
205C3×C3⋊C8—62no—
206#440—271noSylow
207C2×C33C4×D411yescharacteristic
208C2×C33AGL(1,ℤ/8)11yescharacteristic
209C2×C33#13711yescharacteristic
210C3×C3⋊D3—42no—
211C2×C3×Q8—91no—
212C2×C3×D4—91no—
213C22×C3×C4—181no—
214C2×D12—31no—
215C2×D12—31no—
216C2×D12—61no—
217C2×C4×D3—61no—
218C2×C4×D3—61no—
219C2×Dic6—31no—
220C2×Dic6—31no—
221C2×Dic6—61no—
222C2×C3×C8—91no—
223C3×C4⋊C4—31no—
224C3×C4⋊C4—31no—
225C3×C4⋊C4—31no—
226C3×C4⋊C4—31no—
227C3×C4⋊C4—61no—
228C3×C4⋊C4—61no—
229C3×C4⋊C4—181no—
230C3×C22⋊C4—181no—
231C3×C42—31no—
232C3×C42—31no—
233C3×C42—61no—
234C3×C42—181no—
235#215—94no—
236#214—94no—
237(C2×D3)⋊C4—61no—
238(C2×D3)⋊C4—61no—
239C4⋊Dic3D3211yescharacteristic
240C4⋊Dic3D3211yescharacteristic
241Dic3⋊C4—61no—
242Dic3⋊C4—61no—
243C4×Dic3—61no—
244C4×Dic3—61no—
245C4×Dic3D3211yescharacteristic
246C2×C3⋊C8—31no—
247C2×C3⋊C8—31no—
248C2×C3⋊C8—61no—
249C2×C3⋊C8—91no—
250C2×C3⋊C8—91no—
251C2×C3⋊C8—91no—
252C2×C3⋊C8—181no—
253C2×C3⋊C8—181no—
254C2×C3⋊C8—181no—
255C2×C3⋊C8—361no—
256#204—94no—
257C22×C32—21no—
258C22×C32—21no—
259C22×C32—21no—
260C22×C32—41no—
261C22×C32D3×D411yescharacteristic
262C22×C32C2×#6711yescharacteristic
263C22×C32C2×#6711yescharacteristic
264C2×C3⋊D3—22no—
265C2×C3⋊D3—41no—
266C2×C3⋊D3—41no—
267C2×C3×D3—62no—
268C2×C3×D3—62no—
269C2×C3×D3—122no—
270C2×C3×D3—121no—
271C2×C3×D3—121no—
272C2×C3×D3—121no—
273C2×C3×D3—121no—
274C2×C3×D3—241no—
275C2×C3×D3—241no—
276C32×C4—21no—
277C32×C4—21no—
278C32×C4—21no—
279C32×C4—21no—
280C32×C4—21no—
281C32×C4—21no—
282C32×C4—41no—
283C32×C4—41no—
284C32×C4—61no—
285C32×C4—61no—
286C32×C4—121no—
287C32×C4—121no—
288C32×C4C2×C4×D311yescharacteristic
289C32×C4C2×C4×D311yescharacteristic
290C32×C4C2×C4×D311yescharacteristic
291C32×C4#23511yescharacteristic
292C32×C4#23511yescharacteristic
293C32×C4#23911yescharacteristic
294C3⋊Dic3—22no—
295C3×Dic3—21no—
296C3×Dic3—21no—
297C3×Dic3—21no—
298C3×Dic3—21no—
299C3×Dic3—41no—
300C3×Dic3—41no—
301C3×Dic3—41no—
302C3×Dic3—41no—
303C3×Dic3—41no—
304C3×Dic3—41no—
305C3×Dic3—62no—
306C3×Dic3—62no—
307C3×Dic3—81no—
308C3×Dic3—81no—
309C3×Dic3—122no—
310C2×SD16—271no—
311C4×Q8—91no—
312C4×D4—91no—
313#107—91no—
314#103—271no—
315#102—271no—
316#97—271no—
317C33#44011yescharacteristic, complemented, Sylow
318C23×C3—181no—
319C22×D3—61no—
320C22×D3—61no—
321C22×D3—121no—
322C3×Q8—92no—
323C3×Q8—181no—
324C3×D4—92no—
325C3×D4—181no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—31no—
330C2×C3×C4—31no—
331C2×C3×C4—31no—
332C2×C3×C4—31no—
333C2×C3×C4—41no—
334C2×C3×C4—61no—
335C2×C3×C4—61no—
336C2×C3×C4—61no—
337C2×C3×C4—61no—
338C2×C3×C4—61no—
339C2×C3×C4—61no—
340C2×C3×C4—121no—
341C2×C3×C4—121no—
342C2×C3×C4—181no—
343C2×C3×C4—362no—
344C2×C3×C4C2×D3211yescharacteristic
345C2×C3×C4C2×D3211yescharacteristic
346C2×C3×C4C2×D3211yescharacteristic
347C2×Dic3—21no—
348C2×Dic3—21no—
349C2×Dic3—61no—
350C2×Dic3—61no—
351C2×Dic3—121no—
352C2×Dic3#62511yescharacteristic
353C2×Dic3#62511yescharacteristic
354D12—32no—
355D12—32no—
356D12—62no—
357D12—61no—
358D12—61no—
359D12—121no—
360C4×D3—122no—
361C4×D3—122no—
362Dic6—32no—
363Dic6—32no—
364Dic6—62no—
365Dic6—61no—
366Dic6—61no—
367Dic6—121no—
368C3×C8—92no—
369C3⋊C8—32no—
370C3⋊C8—32no—
371C3⋊C8—62no—
372C3⋊C8—182no—
373C3⋊C8—181no—
374C3⋊C8—181no—
375C3⋊C8—181no—
376C3⋊C8—361no—
377C3⋊C8—361no—
378C3⋊C8—362no—
379C2×C32—21no—
380C2×C32—21no—
381C2×C32—21no—
382C2×C32—21no—
383C2×C32—21no—
384C2×C32—21no—
385C2×C32—21no—
386C2×C32—21no—
387C2×C32—21no—
388C2×C32—41no—
389C2×C32—41no—
390C2×C32—41no—
391C2×C32#89311yescharacteristic
392C2×C32#91111yescharacteristic
393C2×C32#91811yescharacteristic
394C2×C32C4×#6711yescharacteristic
395C2×C32C4×#6711yescharacteristic
396C2×C32#93211yescharacteristic
397C2×C32#94211yescharacteristic
398C2×C32#94911yescharacteristic
399C2×C32#95111yescharacteristic
400C3⋊D3—42no—
401C3×D3—122no—
402C3×D3—122no—
403C3×D3—242no—
404C2×Q8—91no—
405C2×D4—91no—
406C22×C4—181no—
407SD16—274no—
408C2×C8—91no—
409C2×C8—271no—
410C4⋊C4—31no—
411C4⋊C4—31no—
412C4⋊C4—181no—
413C22⋊C4—181no—
414C42—31no—
415C42—181no—
416C22×C3—21no—
417C22×C3—21no—
418C22×C3—21no—
419C22×C3—41no—
420C22×C3—182no—
421C22×C3—361no—
422C22×C3—361no—
423C22×C3C2×#62511yescharacteristic
424C22×C3D3×#6711yescharacteristic
425C22×C3D3×#6711yescharacteristic
426C2×D3—62no—
427C2×D3—62no—
428C2×D3—122no—
429C2×D3—121no—
430C2×D3—121no—
431C2×D3—121no—
432C2×D3—121no—
433C2×D3—241no—
434C2×D3—241no—
435C3×C4—21no—
436C3×C4—21no—
437C3×C4—21no—
438C3×C4—21no—
439C3×C4—21no—
440C3×C4—21no—
441C3×C4—41no—
442C3×C4—41no—
443C3×C4—61no—
444C3×C4—61no—
445C3×C4—61no—
446C3×C4—61no—
447C3×C4—121no—
448C3×C4—121no—
449C3×C4—121no—
450C3×C4—121no—
451C3×C4—121no—
452C3×C4—121no—
453C3×C4—182no—
454C3×C4—241no—
455C3×C4—241no—
456C3×C4#379811yescharacteristic
457C3×C4#379911yescharacteristic
458C3×C4#379911yescharacteristic
459C3×C4C4×D3211yescharacteristic
460C3×C4C4×D3211yescharacteristic
461C3×C4C4×D3211yescharacteristic
462Dic3—21no—
463Dic3—21no—
464Dic3—41no—
465Dic3—41no—
466Dic3—62no—
467Dic3—62no—
468Dic3—122no—
469C32—21no—
470C32—21no—
471C32—21no—
472C32—41no—
473C32#488911yescharacteristic, complemented
474C32#501811yescharacteristic, complemented
475C32#502611yescharacteristic, complemented
476C23—181no—
477Q8—92no—
478Q8—181no—
479D4—92no—
480D4—181no—
481C2×C4—31no—
482C2×C4—31no—
483C2×C4—61no—
484C2×C4—61no—
485C2×C4—181no—
486C2×C4—362no—
487C2×C4D3311yescharacteristic
488C8—92no—
489C8—541no—
490C2×C3—21no—
491C2×C3—21no—
492C2×C3—21no—
493C2×C3—21no—
494C2×C3—21no—
495C2×C3—21no—
496C2×C3—21no—
497C2×C3—21no—
498C2×C3—21no—
499C2×C3—41no—
500C2×C3—41no—
501C2×C3—41no—
502C2×C3—362no—
503C2×C3#6382311yescharacteristic
504C2×C3#6382711yescharacteristic
505C2×C3C4×#62511yescharacteristic
506C2×C3#6390411yescharacteristic
507C2×C3#6390411yescharacteristic
508C2×C3#6392511yescharacteristic
509C2×C3#6393211yescharacteristic
510C2×C3#6394311yescharacteristic
511C2×C3#6394811yescharacteristic
512D3—122no—
513D3—122no—
514D3—242no—
515C22—182no—
516C22—361no—
517C22—361no—
518C22D3×#62511yescharacteristic
519C4—61no—
520C4—61no—
521C4—121no—
522C4—121no—
523C4—182no—
524C4#8894911yescharacteristic
525C4#8896511yescharacteristic
526C3—21no—
527C3—21no—
528C3—21no—
529C3—41no—
530C3#1059035811yescharacteristic, complemented
531C3#1059045711yescharacteristic, complemented
532C3#1059059011yescharacteristic, complemented
533C2—362no—
534C2#1172286611yescharacteristic
535C2#1172288311yescharacteristic
536C2#1172293111yescharacteristic
subgroupG/H
centerC22D3×#625
commutatorC33×C4C22×C4
Frattini, hypercenterC2×C4D33
Fitting, cosocleC2×C33×C4C23
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(C3⋊D3)C22
hypocenterC33#440
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. C3×#63844C2semidirect
4. #11722039C2semidirect
6. #11722019C2semidirect
7. C3×#63576C2semidirect
19. #88811C4semidirect
28. #63844D3semidirect
37. #63576D3semidirect
53. C3×#634C2×C4semidirect
60. C3×C3⋊(C3⋊C8)C2×C4semidirect
164. #634C4×D3semidirect
202. C3⋊(C3⋊C8)C4×D3semidirect
317. C33#440semidirect
473. C32#4889semidirect
474. C32#5018semidirect
475. C32#5026semidirect
530. C3#10590358semidirect
531. C3#10590457semidirect
532. C3#10590590semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#888112
C3×#638942
C3×#638442
#117220392
#117220332
#117220192
C3×#635762
#105905903
#105904573
#105903583